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Let (X₁, Y₁), ...,(X_n, Y_n) be i.i.d. random 2-vectors having the bivariate normal distribution with EX₁ = EY₁ = 0, Var(X₁) = σ²_x, Var(Y₁) = σ²_y, and Cov(X₁, Y₁) = ρσ_xσ_y, where σ_x > 0, σ_y > 0, and ρ ∈ [0, 1) are unknown. Derive the form and exact distribution of a UMPU test of size α for testing H₀; ρ = 0 versus H₁; ρ > 0.

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Category:- Statistics and Probability
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